Question

Find the taylor polynomial of order 3 generated by f at a.

f(x)= 1/4-x , a=1

Answer 1

To find the Taylor polynomial of order 3 generated by f at a = 1, we need to find the derivatives of f up to the third order evaluated at a. We can then use these derivatives to construct the polynomial.

Step-by-step explanation:

To find the Taylor polynomial of order 3 generated by f at a, we need to find the derivatives of f up to the third order evaluated at a. Then we can use these derivatives to construct the polynomial.

Given f(x) = 1/(4-x) and a = 1, let's find the derivatives:

1. First derivative: f'(x) = 1/(4-x)^2
2. Second derivative: f''(x) = 2/(4-x)^3
3. Third derivative: f'''(x) = 6/(4-x)^4

Now, let's evaluate these derivatives at a = 1:

1. First derivative at a = 1: f'(1) = 1/(4-1)^2 = 1/9
2. Second derivative at a = 1: f''(1) = 2/(4-1)^3 = 2/27
3. Third derivative at a = 1: f'''(1) = 6/(4-1)^4 = 6/81 = 2/27

Finally, we can use these derivatives to construct the Taylor polynomial of order 3 at a = 1:

T(x) = f(1) + f'(1)(x-1) + f''(1)(x-1)^2 + f'''(1)(x-1)^3

Substituting the values we found:

T(x) = 1/(4-1) + (1/9)(x-1) + (2/27)(x-1)^2 + (2/27)(x-1)^3

Simplifying further will give you the actual polynomial.